\(\int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx\) [538]

Optimal result

Integrand size = 23, antiderivative size = 436 \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {2 \left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (128 a^4-116 a^2 b^2-17 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 b^5 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 a^2 \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {8 a^2 \left (2 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^4 \left (a^2-b^2\right )^2 d}+\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d} \] Output:

2/15*(128*a^6-212*a^4*b^2+55*a^2*b^4+9*b^6)*(a+b*cos(d*x+c))^(1/2)*Ellipti 
cE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/b^5/(a^2-b^2)^2/d/((a+b*cos 
(d*x+c))/(a+b))^(1/2)-2/15*a*(128*a^4-116*a^2*b^2-17*b^4)*((a+b*cos(d*x+c) 
)/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/b^5/ 
(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)-2/3*a^2*cos(d*x+c)^3*sin(d*x+c)/b/(a^2- 
b^2)/d/(a+b*cos(d*x+c))^(3/2)-8/3*a^2*(2*a^2-3*b^2)*cos(d*x+c)^2*sin(d*x+c 
)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(1/2)-4/15*a*(32*a^4-49*a^2*b^2+7*b^4 
)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b^4/(a^2-b^2)^2/d+2/15*(48*a^4-71*a^2* 
b^2+3*b^4)*cos(d*x+c)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b^3/(a^2-b^2)^2/d
 

Mathematica [A] (verified)

Time = 2.53 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.62 \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {\frac {2 \left (\frac {a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (\left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+a \left (-128 a^5+128 a^4 b+116 a^3 b^2-116 a^2 b^3+17 a b^4-17 b^5\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )}{(a-b)^2}+b \left (\frac {10 a^5 \sin (c+d x)}{a^2-b^2}-\frac {10 a^4 \left (11 a^2-15 b^2\right ) (a+b \cos (c+d x)) \sin (c+d x)}{\left (a^2-b^2\right )^2}-28 a (a+b \cos (c+d x))^2 \sin (c+d x)+3 b (a+b \cos (c+d x))^2 \sin (2 (c+d x))\right )}{15 b^5 d (a+b \cos (c+d x))^{3/2}} \] Input:

Integrate[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^(5/2),x]
 

Output:

((2*((a + b*Cos[c + d*x])/(a + b))^(3/2)*((128*a^6 - 212*a^4*b^2 + 55*a^2* 
b^4 + 9*b^6)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + a*(-128*a^5 + 128*a^4 
*b + 116*a^3*b^2 - 116*a^2*b^3 + 17*a*b^4 - 17*b^5)*EllipticF[(c + d*x)/2, 
 (2*b)/(a + b)]))/(a - b)^2 + b*((10*a^5*Sin[c + d*x])/(a^2 - b^2) - (10*a 
^4*(11*a^2 - 15*b^2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2 - 28 
*a*(a + b*Cos[c + d*x])^2*Sin[c + d*x] + 3*b*(a + b*Cos[c + d*x])^2*Sin[2* 
(c + d*x)]))/(15*b^5*d*(a + b*Cos[c + d*x])^(3/2))
 

Rubi [A] (verified)

Time = 2.55 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.01, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 3271, 27, 3042, 3526, 27, 3042, 3528, 27, 3042, 3502, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^5}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\)

\(\Big \downarrow \) 3271

\(\displaystyle -\frac {2 \int \frac {\cos ^2(c+d x) \left (6 a^2-3 b \cos (c+d x) a-\left (8 a^2-3 b^2\right ) \cos ^2(c+d x)\right )}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) \left (6 a^2-3 b \cos (c+d x) a-\left (8 a^2-3 b^2\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (6 a^2-3 b \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (3 b^2-8 a^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3526

\(\displaystyle -\frac {\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \int -\frac {\cos (c+d x) \left (16 \left (2 a^2-3 b^2\right ) a^2-2 b \left (a^2-3 b^2\right ) \cos (c+d x) a-\left (48 a^4-71 b^2 a^2+3 b^4\right ) \cos ^2(c+d x)\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\int \frac {\cos (c+d x) \left (16 \left (2 a^2-3 b^2\right ) a^2-2 b \left (a^2-3 b^2\right ) \cos (c+d x) a-\left (48 a^4-71 b^2 a^2+3 b^4\right ) \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (16 \left (2 a^2-3 b^2\right ) a^2-2 b \left (a^2-3 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (-48 a^4+71 b^2 a^2-3 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3528

\(\displaystyle -\frac {\frac {\frac {2 \int -\frac {-6 a \left (32 a^4-49 b^2 a^2+7 b^4\right ) \cos ^2(c+d x)-b \left (16 a^4-27 b^2 a^2-9 b^4\right ) \cos (c+d x)+2 a \left (48 a^4-71 b^2 a^2+3 b^4\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {-\frac {\int \frac {-6 a \left (32 a^4-49 b^2 a^2+7 b^4\right ) \cos ^2(c+d x)-b \left (16 a^4-27 b^2 a^2-9 b^4\right ) \cos (c+d x)+2 a \left (48 a^4-71 b^2 a^2+3 b^4\right )}{\sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {-\frac {\int \frac {-6 a \left (32 a^4-49 b^2 a^2+7 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (16 a^4-27 b^2 a^2-9 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a \left (48 a^4-71 b^2 a^2+3 b^4\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3502

\(\displaystyle -\frac {\frac {-\frac {\frac {2 \int \frac {3 \left (4 a b \left (8 a^4-11 b^2 a^2-2 b^4\right )+\left (128 a^6-212 b^2 a^4+55 b^4 a^2+9 b^6\right ) \cos (c+d x)\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {-\frac {\frac {\int \frac {4 a b \left (8 a^4-11 b^2 a^2-2 b^4\right )+\left (128 a^6-212 b^2 a^4+55 b^4 a^2+9 b^6\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {-\frac {\frac {\int \frac {4 a b \left (8 a^4-11 b^2 a^2-2 b^4\right )+\left (128 a^6-212 b^2 a^4+55 b^4 a^2+9 b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3231

\(\displaystyle -\frac {\frac {-\frac {\frac {\frac {\left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {a \left (128 a^6-244 a^4 b^2+99 a^2 b^4+17 b^6\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{b}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {-\frac {\frac {\frac {\left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {a \left (128 a^6-244 a^4 b^2+99 a^2 b^4+17 b^6\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {\frac {-\frac {\frac {\frac {\left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (128 a^6-244 a^4 b^2+99 a^2 b^4+17 b^6\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {-\frac {\frac {\frac {\left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (128 a^6-244 a^4 b^2+99 a^2 b^4+17 b^6\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {\frac {-\frac {\frac {\frac {2 \left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (128 a^6-244 a^4 b^2+99 a^2 b^4+17 b^6\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {\frac {-\frac {\frac {\frac {2 \left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (128 a^6-244 a^4 b^2+99 a^2 b^4+17 b^6\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{b}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {-\frac {\frac {\frac {2 \left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (128 a^6-244 a^4 b^2+99 a^2 b^4+17 b^6\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{b}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {\frac {8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {-\frac {2 \left (48 a^4-71 a^2 b^2+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}-\frac {\frac {\frac {2 \left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (128 a^6-244 a^4 b^2+99 a^2 b^4+17 b^6\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {4 a \left (32 a^4-49 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}}{b \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\)

Input:

Int[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^(5/2),x]
 

Output:

(-2*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x 
])^(3/2)) - ((8*a^2*(2*a^2 - 3*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - 
 b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((-2*(48*a^4 - 71*a^2*b^2 + 3*b^4)*Cos 
[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d) - (((2*(128*a^6 - 
 212*a^4*b^2 + 55*a^2*b^4 + 9*b^6)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + 
 d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*( 
128*a^6 - 244*a^4*b^2 + 99*a^2*b^4 + 17*b^6)*Sqrt[(a + b*Cos[c + d*x])/(a 
+ b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]] 
))/b - (4*a*(32*a^4 - 49*a^2*b^2 + 7*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + 
 d*x])/(b*d))/(5*b))/(b*(a^2 - b^2)))/(3*b*(a^2 - b^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[d/b   Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b 
, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
 

rule 3271

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co 
s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* 
(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin 
[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 
2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 
+ b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - 
 d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] 
, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - 
b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || 
IntegersQ[2*m, 2*n])
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

rule 3526

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - 
d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin[e + f*x])^(m - 
 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* 
d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 
1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x 
] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f 
*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d 
, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
 

rule 3528

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ 
.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x 
])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} 
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ 
m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1687\) vs. \(2(417)=834\).

Time = 10.68 (sec) , antiderivative size = 1688, normalized size of antiderivative = 3.87

method	result	size
default	\(\text {Expression too large to display}\)	\(1688\)

Input:

int(cos(d*x+c)^5/(a+cos(d*x+c)*b)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(16/b^2*(-1 
/10/b*cos(1/2*d*x+1/2*c)^3*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/ 
2*c)^2)^(1/2)-1/60/b^2*(-4*a+12*b)*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/ 
2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+1/60/b^2*(-4*a+12*b)*(a-b)*(sin(1 
/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b* 
sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d 
*x+1/2*c),(-2*b/(a-b))^(1/2))-1/60*(4*a^2-15*a*b+27*b^2)/b^3*(a-b)*(sin(1/ 
2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*s 
in(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d 
*x+1/2*c),(-2*b/(a-b))^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1 
/2))))-2*(4*a^3+3*a^2*b+2*a*b^2+b^3)/b^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2* 
b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)* 
sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2 
))-10/b^5*a^4/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2*b-a-b)/(a^2-b^2 
)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*cos(1/2* 
d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*b+(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b) 
*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2* 
b/(a-b))^(1/2))*a-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2 
*c)^2+(a+b)/(a-b))^(1/2)*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2) 
))-2/b^5*a^5*(1/6/b/(a-b)/(a+b)*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 1008, normalized size of antiderivative = 2.31 \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

-2/45*(2*sqrt(1/2)*(-128*I*a^9 + 260*I*a^7*b^2 - 121*I*a^5*b^4 - 21*I*a^3* 
b^6 + (-128*I*a^7*b^2 + 260*I*a^5*b^4 - 121*I*a^3*b^6 - 21*I*a*b^8)*cos(d* 
x + c)^2 + 2*(-128*I*a^8*b + 260*I*a^6*b^3 - 121*I*a^4*b^5 - 21*I*a^2*b^7) 
*cos(d*x + c))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27* 
(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b 
) + 2*sqrt(1/2)*(128*I*a^9 - 260*I*a^7*b^2 + 121*I*a^5*b^4 + 21*I*a^3*b^6 
+ (128*I*a^7*b^2 - 260*I*a^5*b^4 + 121*I*a^3*b^6 + 21*I*a*b^8)*cos(d*x + c 
)^2 + 2*(128*I*a^8*b - 260*I*a^6*b^3 + 121*I*a^4*b^5 + 21*I*a^2*b^7)*cos(d 
*x + c))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 
 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3* 
sqrt(1/2)*(-128*I*a^8*b + 212*I*a^6*b^3 - 55*I*a^4*b^5 - 9*I*a^2*b^7 + (-1 
28*I*a^6*b^3 + 212*I*a^4*b^5 - 55*I*a^2*b^7 - 9*I*b^9)*cos(d*x + c)^2 + 2* 
(-128*I*a^7*b^2 + 212*I*a^5*b^4 - 55*I*a^3*b^6 - 9*I*a*b^8)*cos(d*x + c))* 
sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b 
^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b 
^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)) + 3*sqrt(1/2)*(1 
28*I*a^8*b - 212*I*a^6*b^3 + 55*I*a^4*b^5 + 9*I*a^2*b^7 + (128*I*a^6*b^3 - 
 212*I*a^4*b^5 + 55*I*a^2*b^7 + 9*I*b^9)*cos(d*x + c)^2 + 2*(128*I*a^7*b^2 
 - 212*I*a^5*b^4 + 55*I*a^3*b^6 + 9*I*a*b^8)*cos(d*x + c))*sqrt(b)*weierst 
rassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstr...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**5/(a+b*cos(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{5}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

integrate(cos(d*x + c)^5/(b*cos(d*x + c) + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{5}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

integrate(cos(d*x + c)^5/(b*cos(d*x + c) + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^5}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:

int(cos(c + d*x)^5/(a + b*cos(c + d*x))^(5/2),x)
 

Output:

int(cos(c + d*x)^5/(a + b*cos(c + d*x))^(5/2), x)
 

Reduce [F]

\[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{5}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \] Input:

int(cos(d*x+c)^5/(a+b*cos(d*x+c))^(5/2),x)
 

Output:

int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**5)/(cos(c + d*x)**3*b**3 + 3*c 
os(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)